426 APPENDIX. 



vector, and another in a line perpendicular to that radius 

 vector. Thus Clairaut (Mem. Acad. 1748, p. 435.) gives 

 these equations r d 2 v -{- 2 drdv = II d x~ 



r being the radius vector, v its angle with the axis, dx the 

 differential of the time, II the force to the centre, S the 

 disturbing force. So D Alembert (Mem. Acad. 1745, 

 p. 365.) takes the same course, and obtains an equation 

 to the orbit in question, depending on the integration of 



dz, TI being the disturbing force acting in a line 



perpendicular to the radius vector, and z the circular arc 

 described with a radius equal to the distance between the 

 centre of force and the vertex of the orbit. This assumes, 

 however, that the orbit is itself nearly circular. 



4. If P = distance of E (Earth) from Moon (M) s 

 quadrature, s = sin. angle of rad. vec. r with the per 

 pendicular to a, the distance of E from S, the Sun; 



i -A r AT Ai j rdP SP^mnsds 



v = velocity of M ; then v d v = p -- h - 3 



supposing the motion of M to be almost uniform. Here 

 one of the forces acting on M is directed towards E and 



E 4- M S x M E . 

 is = , , p 2 - + , , 3 ; the other force is in a line 



parallel to S E or a, and is = , _ 3 . It was 



in consequence of this investigation that Clairaut for some 

 time announced, as did also Euler and D Alembert, that 

 there was a material error in the Newtonian theory of the 

 Moon s motion. The error, which afterwards was found 

 to arise from their having omitted the consideration of 

 certain quantities, was acknowledged by Clairaut three 

 years later (Mem. Acad. 1748, pp. 421. 434.), but no one 



